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Integral
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[tex]\int \frac{3 x^{2} + x + 4}{x^{4} + 3 x^{2} + 2}\, dx = \frac{1}{2} \log{\left (x^{2} + 1 \right )} - \frac{1}{2} \log{\left (x^{2} + 2 \right )} + \operatorname{atan}{\left (x \right )} + \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )} + \mathrm{const}[/tex]

Integral Steps:

  1. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

      [tex]\frac{3 x^{2} + x + 4}{x^{4} + 3 x^{2} + 2} = - \frac{x - 2}{x^{2} + 2} + \frac{x + 1}{x^{2} + 1}[/tex]

    2. Integrate term-by-term:

      1. The integral of a constant times a function is the constant times the integral of the function:

        [tex]\int - \frac{x - 2}{x^{2} + 2}\, dx = - \int \frac{x - 2}{x^{2} + 2}\, dx[/tex]

        1. Rewrite the integrand:

          [tex]\frac{x - 2}{x^{2} + 2} = \frac{x}{x^{2} + 2} - \frac{2}{x^{2} + 2}[/tex]

        2. Integrate term-by-term:

          1. Let [tex]u = x^{2} + 2[/tex].

            Then let [tex]du = 2 x dx[/tex] and substitute [tex]\frac{du}{2}[/tex]:

            [tex]\int \frac{1}{2 u}\, du[/tex]

            1. The integral of a constant times a function is the constant times the integral of the function:

              [tex]\int \frac{1}{u}\, du = \frac{1}{2} \int \frac{1}{u}\, du[/tex]

              1. The result is: [tex]\log{u}[/tex]

              So, the result is: [tex]\frac{1}{2} \log{\left (u \right )}[/tex]

            Now substitute [tex]u[/tex] back in:

            [tex]\frac{1}{2} \log{\left (x^{2} + 2 \right )}[/tex]

          1. The integral of a constant times a function is the constant times the integral of the function:

            [tex]\int - \frac{2}{x^{2} + 2}\, dx = - 2 \int \frac{1}{x^{2} + 2}\, dx[/tex]

            1. The integral of a constant times a function is the constant times the integral of the function:

              [tex]\int \frac{1}{x^{2} + 2}\, dx = \frac{1}{2} \int \frac{1}{\frac{x^{2}}{2} + 1}\, dx[/tex]

              1. Let [tex]u = \frac{\sqrt{2} x}{2}[/tex].

                Then let [tex]du = \frac{\sqrt{2} dx}{2}[/tex] and substitute [tex]\sqrt{2} du[/tex]:

                [tex]\int \frac{2}{u^{2} + 1}\, du[/tex]

                1. The integral of a constant times a function is the constant times the integral of the function:

                  [tex]\int \frac{\sqrt{2}}{u^{2} + 1}\, du = \sqrt{2} \int \frac{1}{u^{2} + 1}\, du[/tex]

                  1. The integral of [tex]\frac{1}{u^{2} + 1}[/tex] is [tex]\operatorname{atan}{\left (u \right )}[/tex].

                  So, the result is: [tex]\sqrt{2} \operatorname{atan}{\left (u \right )}[/tex]

                Now substitute [tex]u[/tex] back in:

                [tex]\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}[/tex]

              So, the result is: [tex]\frac{\sqrt{2}}{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}[/tex]

            So, the result is: [tex]- \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}[/tex]

          The result is: [tex]\frac{1}{2} \log{\left (x^{2} + 2 \right )} - \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}[/tex]

        So, the result is: [tex]- \frac{1}{2} \log{\left (x^{2} + 2 \right )} + \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}[/tex]

      1. Rewrite the integrand:

        [tex]\frac{x + 1}{x^{2} + 1} = \frac{x}{x^{2} + 1} + \frac{1}{x^{2} + 1}[/tex]

      2. Integrate term-by-term:

        1. Let [tex]u = x^{2} + 1[/tex].

          Then let [tex]du = 2 x dx[/tex] and substitute [tex]\frac{du}{2}[/tex]:

          [tex]\int \frac{1}{2 u}\, du[/tex]

          1. The integral of a constant times a function is the constant times the integral of the function:

            [tex]\int \frac{1}{u}\, du = \frac{1}{2} \int \frac{1}{u}\, du[/tex]

            1. The result is: [tex]\log{u}[/tex]

            So, the result is: [tex]\frac{1}{2} \log{\left (u \right )}[/tex]

          Now substitute [tex]u[/tex] back in:

          [tex]\frac{1}{2} \log{\left (x^{2} + 1 \right )}[/tex]

        1. The integral of [tex]\frac{1}{x^{2} + 1}[/tex] is [tex]\operatorname{atan}{\left (x \right )}[/tex].

        The result is: [tex]\frac{1}{2} \log{\left (x^{2} + 1 \right )} + \operatorname{atan}{\left (x \right )}[/tex]

      The result is: [tex]\frac{1}{2} \log{\left (x^{2} + 1 \right )} - \frac{1}{2} \log{\left (x^{2} + 2 \right )} + \operatorname{atan}{\left (x \right )} + \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}[/tex]

    Method #2

    1. Rewrite the integrand:

      [tex]\frac{3 x^{2} + x + 4}{x^{4} + 3 x^{2} + 2} = \frac{3 x^{2}}{x^{4} + 3 x^{2} + 2} + \frac{x}{x^{4} + 3 x^{2} + 2} + \frac{4}{x^{4} + 3 x^{2} + 2}[/tex]

    2. Integrate term-by-term:

      1. The integral of a constant times a function is the constant times the integral of the function:

        [tex]\int \frac{3 x^{2}}{x^{4} + 3 x^{2} + 2}\, dx = 3 \int \frac{x^{2}}{x^{4} + 3 x^{2} + 2}\, dx[/tex]

        1. Rewrite the integrand:

          [tex]\frac{x^{2}}{x^{4} + 3 x^{2} + 2} = \frac{2}{x^{2} + 2} - \frac{1}{x^{2} + 1}[/tex]

        2. Integrate term-by-term:

          1. The integral of a constant times a function is the constant times the integral of the function:

            [tex]\int \frac{2}{x^{2} + 2}\, dx = 2 \int \frac{1}{x^{2} + 2}\, dx[/tex]

            1. The integral of a constant times a function is the constant times the integral of the function:

              [tex]\int \frac{1}{x^{2} + 2}\, dx = \frac{1}{2} \int \frac{1}{\frac{x^{2}}{2} + 1}\, dx[/tex]

              1. Let [tex]u = \frac{\sqrt{2} x}{2}[/tex].

                Then let [tex]du = \frac{\sqrt{2} dx}{2}[/tex] and substitute [tex]\sqrt{2} du[/tex]:

                [tex]\int \frac{2}{u^{2} + 1}\, du[/tex]

                1. The integral of a constant times a function is the constant times the integral of the function:

                  [tex]\int \frac{\sqrt{2}}{u^{2} + 1}\, du = \sqrt{2} \int \frac{1}{u^{2} + 1}\, du[/tex]

                  1. The integral of [tex]\frac{1}{u^{2} + 1}[/tex] is [tex]\operatorname{atan}{\left (u \right )}[/tex].

                  So, the result is: [tex]\sqrt{2} \operatorname{atan}{\left (u \right )}[/tex]

                Now substitute [tex]u[/tex] back in:

                [tex]\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}[/tex]

              So, the result is: [tex]\frac{\sqrt{2}}{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}[/tex]

            So, the result is: [tex]\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}[/tex]

          1. The integral of a constant times a function is the constant times the integral of the function:

            [tex]\int - \frac{1}{x^{2} + 1}\, dx = - \int \frac{1}{x^{2} + 1}\, dx[/tex]

            1. The integral of [tex]\frac{1}{x^{2} + 1}[/tex] is [tex]\operatorname{atan}{\left (x \right )}[/tex].

            So, the result is: [tex]- \operatorname{atan}{\left (x \right )}[/tex]

          The result is: [tex]- \operatorname{atan}{\left (x \right )} + \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}[/tex]

        So, the result is: [tex]- 3 \operatorname{atan}{\left (x \right )} + 3 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}[/tex]

      1. Rewrite the integrand:

        [tex]\frac{x}{x^{4} + 3 x^{2} + 2} = - \frac{x}{x^{2} + 2} + \frac{x}{x^{2} + 1}[/tex]

      2. Integrate term-by-term:

        1. The integral of a constant times a function is the constant times the integral of the function:

          [tex]\int - \frac{x}{x^{2} + 2}\, dx = - \int \frac{x}{x^{2} + 2}\, dx[/tex]

          1. Let [tex]u = x^{2} + 2[/tex].

            Then let [tex]du = 2 x dx[/tex] and substitute [tex]\frac{du}{2}[/tex]:

            [tex]\int \frac{1}{2 u}\, du[/tex]

            1. The integral of a constant times a function is the constant times the integral of the function:

              [tex]\int \frac{1}{u}\, du = \frac{1}{2} \int \frac{1}{u}\, du[/tex]

              1. The result is: [tex]\log{u}[/tex]

              So, the result is: [tex]\frac{1}{2} \log{\left (u \right )}[/tex]

            Now substitute [tex]u[/tex] back in:

            [tex]\frac{1}{2} \log{\left (x^{2} + 2 \right )}[/tex]

          So, the result is: [tex]- \frac{1}{2} \log{\left (x^{2} + 2 \right )}[/tex]

        1. Let [tex]u = x^{2} + 1[/tex].

          Then let [tex]du = 2 x dx[/tex] and substitute [tex]\frac{du}{2}[/tex]:

          [tex]\int \frac{1}{2 u}\, du[/tex]

          1. The integral of a constant times a function is the constant times the integral of the function:

            [tex]\int \frac{1}{u}\, du = \frac{1}{2} \int \frac{1}{u}\, du[/tex]

            1. The result is: [tex]\log{u}[/tex]

            So, the result is: [tex]\frac{1}{2} \log{\left (u \right )}[/tex]

          Now substitute [tex]u[/tex] back in:

          [tex]\frac{1}{2} \log{\left (x^{2} + 1 \right )}[/tex]

        The result is: [tex]\frac{1}{2} \log{\left (x^{2} + 1 \right )} - \frac{1}{2} \log{\left (x^{2} + 2 \right )}[/tex]

      1. The integral of a constant times a function is the constant times the integral of the function:

        [tex]\int \frac{4}{x^{4} + 3 x^{2} + 2}\, dx = 4 \int \frac{1}{x^{4} + 3 x^{2} + 2}\, dx[/tex]

        1. Rewrite the integrand:

          [tex]\frac{1}{x^{4} + 3 x^{2} + 2} = - \frac{1}{x^{2} + 2} + \frac{1}{x^{2} + 1}[/tex]

        2. Integrate term-by-term:

          1. The integral of a constant times a function is the constant times the integral of the function:

            [tex]\int - \frac{1}{x^{2} + 2}\, dx = - \int \frac{1}{x^{2} + 2}\, dx[/tex]

            1. The integral of a constant times a function is the constant times the integral of the function:

              [tex]\int \frac{1}{x^{2} + 2}\, dx = \frac{1}{2} \int \frac{1}{\frac{x^{2}}{2} + 1}\, dx[/tex]

              1. Let [tex]u = \frac{\sqrt{2} x}{2}[/tex].

                Then let [tex]du = \frac{\sqrt{2} dx}{2}[/tex] and substitute [tex]\sqrt{2} du[/tex]:

                [tex]\int \frac{2}{u^{2} + 1}\, du[/tex]

                1. The integral of a constant times a function is the constant times the integral of the function:

                  [tex]\int \frac{\sqrt{2}}{u^{2} + 1}\, du = \sqrt{2} \int \frac{1}{u^{2} + 1}\, du[/tex]

                  1. The integral of [tex]\frac{1}{u^{2} + 1}[/tex] is [tex]\operatorname{atan}{\left (u \right )}[/tex].

                  So, the result is: [tex]\sqrt{2} \operatorname{atan}{\left (u \right )}[/tex]

                Now substitute [tex]u[/tex] back in:

                [tex]\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}[/tex]

              So, the result is: [tex]\frac{\sqrt{2}}{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}[/tex]

            So, the result is: [tex]- \frac{\sqrt{2}}{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}[/tex]

          1. The integral of [tex]\frac{1}{x^{2} + 1}[/tex] is [tex]\operatorname{atan}{\left (x \right )}[/tex].

          The result is: [tex]\operatorname{atan}{\left (x \right )} - \frac{\sqrt{2}}{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}[/tex]

        So, the result is: [tex]4 \operatorname{atan}{\left (x \right )} - 2 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}[/tex]

      The result is: [tex]\frac{1}{2} \log{\left (x^{2} + 1 \right )} - \frac{1}{2} \log{\left (x^{2} + 2 \right )} + \operatorname{atan}{\left (x \right )} + \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}[/tex]

  2. Add the constant of integration:

    [tex]\frac{1}{2} \log{\left (x^{2} + 1 \right )} - \frac{1}{2} \log{\left (x^{2} + 2 \right )} + \operatorname{atan}{\left (x \right )} + \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}+ \mathrm{constant}[/tex]

The answer is:

[tex]\frac{1}{2} \log{\left (x^{2} + 1 \right )} - \frac{1}{2} \log{\left (x^{2} + 2 \right )} + \operatorname{atan}{\left (x \right )} + \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}+ \mathrm{constant}[/tex]

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Common Integrals

$\int 0dx = \text{const}$

$\int\ dx = x + \text{const}$

$\int kdx = kx + \text{const}$

$\int x^n\ dx = \frac{1}{n+1}x^{n+1} + \text{const}$ here n≠-1

$\int \frac{1}{x}\ dx = \int x^{-1}\ dx = \ln|x| + \text{const}$

$\int x^{-n}\ dx = \frac{1}{-n+1}x^{-n+1} + \text{const}$

$\int \frac{1}{ax+b}\ dx = \frac{1}{a}\ln|ax+b| + \text{const}$

$\int e^x\ dx = e^x + \text{const}$

$\int a^x\ dx = \frac{a^x}{\\ln a} + \text{const}$

$\int \sin(x)\ dx = -\cos(x) + \text{const}$

$\int \cos(x)\ dx = \sin(x) + \text{const}$

$\int \tan(x)\ dx = \ln|sec(x)| + \text{const}$

$\int \cot(x)\ dx = \ln|\sin(x)| + \text{const}$

$\int \frac{1}{\sqrt{1-x^2}} \ dx = \arcsin(x) + \text{const}$

$\int -\frac{1}{\sqrt{1-x^2}} \ dx = \arccos(x) + \text{const}$

$\int \frac{1}{1+ x^2}\ dx = \arctan(x) + \text{const}$

$\int -\frac{1}{1+x^2}\ dx = \text{arccot}(x) + \text{const}$

Integration by Parts

$\int u\ dv = uv - \int v\ du$

$\int\limits_{a}^{b} u\ dv = uv |_a^b - \int v\ du$

Trigonometric Substitutions

$\sqrt{a^2 - b^2x^2}$ $\Rightarrow x=\frac{a}{b}\sin\theta$ and $\cos^2\theta = 1 - \sin^2\theta$

$\sqrt{a^2 + b^2x^2}$ $\Rightarrow x=\frac{a}{b}\tan\theta$ and $\sec^2\theta = 1 + \tan^2\theta$

$\sqrt{b^2x^2 - a^2}$ $\Rightarrow x=\frac{a}{b}\sec\theta$ and $\tan^2\theta = \sec^2\theta - 1$

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